How do you find the LCM of #5y^4+30y^3-35y^2# and #7y^8+98y^7+343y^6#?

1 Answer
George C.
Sep 25, 2017

#35y^9+455y^8+1225y^7-1715y^6#

Explanation:

Given:

#5y^4+30y^3-35y^2#

#7y^8+98y^7+343y^6#

The least common multiple (LCM) is the product, divided by the greatest common factor (GCF).

Let us start by factoring the two polynomials:

#5y^4+30y^3-35y^2 = 5y^2(y^2+6y-7) = 5y^2(y+7)(y-1)#

#7y^8+98y^7+343y^6 = 7y^6(y^2+14y+49) = 7y^6(y+7)(y+7)#

So the GCF of these two polynomials is #y^2(y+7)# and their LCM is:

#((5y^2(y+7)(y-1))(7y^6(y+7)(y+7)))/(y^2(y+7))#

#= 35y^6(y+7)^2(y-1)#

#= 35y^6(y^2+14y+49)(y-1)#

#= 35y^6(y^3+13y^2+35y-49)#

#= 35y^9+455y^8+1225y^7-1715y^6#

Note that instead of dividing the product of the polynomials by their GCF, we can simply form the product of each of the factors that occur in either of the polynomials, with sufficient multiplicity.

However, it may be better to find the GCF, since we do not actually have to factor the polynomials to do that.

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